States of a spin 1/2 particle in a magnetic field

[ulcrid]

Hi, for my quantum mechanics course I'm confronted with the following question:

Homework Statement

A spin 1/2 particle has the following eigenstates of S_z: |+> = (1 0) and |-> = (0 1). A magnetic field is pointing in the z direction, B = (0,0,B). The Hamiltonian is H = -B * n, with n = -(e/mc)S and S = h/4pi * s (with B, n, S, and s vectors and H of course an operator).

The questions are a) to find the normalized energy eigenstates and eigenvalues and b) to find the normalized eigenstates and eigenvalues of S_x in terms of the eigenstates of S_z.

Homework Equations

The Pauli spin matrices are most important here, for a) only the z component is relevant as the magnetic field only has a z component and the Hamiltonian is defined as -B * n. For b) I think the x component is also relevant.
s_z=
(1 0)
(0 -1)
s_x=
(1 0)
(0 1)

The Attempt at a Solution

I think I got the most far on question a). I calculated H and found it to be H = (ehB/4pi*mc)*
(1 0)
(0 -1)
Then, using the determinant of (H - lambda * I) I calculated the eigenvalues of H: lambda = +/- (ehB/4pi*mc). After that, I used the eigenvalue equation Av = lambda*v to find the eigenvectors (1 0) and (0 1). I don't think that should be surprising though because somewhere in the book (Introductory Quantum Mechanics by Liboff) it says that S_z and H have the same eigenfunctions.
But are these normalized? And are these values/vectors correct?

On question b) I don't really know what to do - if I try finding out the eigenstates of S_x I get stuck on the vectors. I get equations like h/4pi (b a) = (a b) which doesn't have a solution except for a = b = 0 which obviously is incorrect. How do I find the correct eigenstates? And how do I then find out how to write them as combinations of eigenstates of S_z? Hope someone can help!

 

[vela]

The Attempt at a Solution

I think I got the most far on question a). I calculated H and found it to be H = (ehB/4pi*mc)*
(1 0)
(0 -1)
Then, using the determinant of (H - lambda * I) I calculated the eigenvalues of H: lambda = +/- (ehB/4pi*mc). After that, I used the eigenvalue equation Av = lambda*v to find the eigenvectors (1 0) and (0 1). I don't think that should be surprising though because somewhere in the book (Introductory Quantum Mechanics by Liboff) it says that S_z and H have the same eigenfunctions.
But are these normalized? And are these values/vectors correct?

If the vectors are normalized, they'd have a magnitude of 1. What are the magnitude of your vectors?

On question b) I don't really know what to do - if I try finding out the eigenstates of S_x I get stuck on the vectors. I get equations like h/4pi (b a) = (a b) which doesn't have a solution except for a = b = 0 which obviously is incorrect. How do I find the correct eigenstates? And how do I then find out how to write them as combinations of eigenstates of S_z? Hope someone can help!

You need to show more details of your work. You might just be dropping a constant somewhere.

The matrix representation of S_x you're using is with respect to basis of the S_z eigenstates, so the eigenvectors you find will already be representations in terms of the eigenstates of S_z.

 

[ulcrid]

Well, the magnitude of the vectors seems to be 1. After all, |(1 0)|² is 1. So that would be okay?

For the second question I used the Pauli-spinmatrix of S_x:

[U]h[/U]   (0 1)
4pi (1 0)

Then, I can use det(S_x-lambda*I) = 0 to get lambda²-1 = 0 so lambda is either 1 or -1.
Using that in the eigenvalue equation I get S_x*v = +/- v
Which gets me to

[U]h[/U]   (0 1)(a) = +/- (a)
4pi (1 0)(b)       (b)

Which is equal to

[U]h[/U]   (b) = +/- (a)
4pi (a)       (b)

Which in my view is only solvable for a = b = 0. ?

 

[vela]

Your eigenvalues should be $\pm\hbar/2$. Can you see the mistake you made in calculating them? If you fix those, the eigenvectors should work out for you.

 

[ulcrid]

Ah, yes, I see where I went wrong. And because I used the Pauli-spinmatrix for S_x, which apparently already is in the S_z/S² representation, these vectors and eigenvectors are already in terms of the eigenstates of S_z?